Wiring a ceiling fan

I think you have a hot-neutral reversal somewhere. You don't show it, but I'm going to assume that you have a ground wire in there somewhere. If you have a multimeter, the voltage between white and ground should be 0 (or almost 0, a couple of volts at most). If not, then you need to take apart all of the connections in the circuit and test each one to figure out where the reversal was introduced and fix it.

You can't have one switch operate light OR fan. Many jurisdictions absolutely require that a switch on the wall near a room entrance, control a lamp. That is to benefit house guests and mainly from the government's perspective, first responders. There's an exception to allow a switched outlet instead, presuming a person is going to plug a floor lamp into that. There is no exception to allow a fan/light locally switched at the fan. (such things can exist, they can't be the only light). That said:

They now make gadgets to solve the "I have 14/2 and wish I had 14/3" problem. Each has a smart switch (typically 1-gang) with 2 controls, and a control module which sits under the fan shroud. The switch communicates with the module either via powerline networking or via wireless, and some of those are compatible with smartphones or integrated home-automation systems as are now emerging in the market.